Locomotion and fluid pumping near surfaces are ubiquitous in nature, ranging from the slow crawling of snails to the rapid flight of bats. This study categorizes these behaviors based on the Undulation number ($\text{Un}$) and Reynolds number ($Re$). We contrast low $Re$ undulatory propulsion ($\text{Un} > 1$), exemplified by freshwater snails, with high $Re$ flapping propulsion ($\text{Un} < 1$), seen in bats and bees. For snails, we derive lubrication models showing that pumping and swimming speeds scale with $(a/h_0)^2$, a result validated by robotic experiments which also reveal the detrimental effects of surface deformation (high Capillary/Bond ratio). Conversely, for high $Re$ fliers, we examine the ground effect's role in lift enhancement. Biological data from bats (\textit{R. ferrumequinum}) reveal a 2.5-fold increase in lift coefficient during surface-skimming drinking flights, attributed to aerodynamic squeezing effects. Finally, we analyze honeybee fanning, demonstrating how a "jet-vortex" mechanism utilizes ground effect to transport pheromones efficiently against diffusion. These findings provide a unified framework for understanding fluid-structure interactions near boundaries in biological systems.
However, the physical mechanisms governing these interactions vary drastically depending on the scale of the organism and the nature of the movement. While the aerodynamic ground effect for high-speed fliers is well-characterized by potential flow and vortex dynamics [12], the hydrodynamics of slow-moving organisms near deformable interfaces remain less understood. For instance, freshwater snails utilize a muscular foot to generate localized currents [13,14], operating in a regime where viscosity dominates and the free surface can deform. Conversely, flapping fliers like bees and bats operate in inertial regimes where unsteady effects, such as vortex shedding and air compression, become significant.
In this manuscript, we present a comparative framework for animal interaction with surfaces, categorizing behaviors based on the Reynolds number (Re) and the Undulation number (Un). We contrast two distinct regimes: (1) Low Reynolds number undulation (Un > 1), exemplified by freshwater snails. We derive lubrication models to show how pumping and swimming speeds scale with the square of the amplitude-to-gap ratio (a/h 0 ) 2 and investigate the efficiency losses caused by surface deformation (Ca/Bo ≫ 1). ( 2) High Reynolds number flapping (Un < 1), represented by bats and bees. We examine the “drinking on the wing” behavior in bats, revealing a 2.5-fold lift enhancement driven by the aerodynamic squeezing effect. Finally, we analyze honeybee fanning, demonstrating how a jet-vortex mechanism utilizes the ground effect to transport pheromones efficiently against diffusion. These diverse examples illustrate how animals leverage distinct physical principles to master the ground effect for survival.
To classify animal movements or pumping, we utilize the Undulation number (Un), defined as the ratio of the active body length (L active ) to the undulation wavelength (λ). This dimensionless number provides a continuum for categorizing propulsion/pumping strategies across different fluid media. Biological observations reveal a wide range of animals that undulate their entire bodies and those that rely on oscillating appendages [16]. In the aquatic realm, undulatory modes are traditionally classified by the fraction of the body participating in the wave motion. At one end of the spectrum lies the Anguilliform mode (Un ≈ 1.0), exemplified by eels and lampreys [17]. In this mode, the entire body undulates, with more than one full wave fitting along the body length (Un > 1). While highly efficient in vis- cous or granular media, this whole-body undulation becomes less efficient at high swimming speeds. As we move towards faster swimmers, the undulation becomes increasingly confined to the posterior. In the Thunniform mode (Un ≈ 0.5), seen in tuna and sharks, movement is concentrated almost entirely in the caudal fin. Here, the tail acts as a rigid hydrofoil pivoting to generate thrust/lift, offering high efficiency for sustained high-speed swimming [18]. The extreme case is the Ostraciiform mode (Un ≈ 0.25), where propulsion is generated solely by the oscillation of a rigid tail fin, as seen in boxfish.
In contrast to aquatic undulators, aerial fliers operate in the “Flapping” regime where the Undulation number is significantly less than unity (Un < 1). The mechanism here shift fundamentally from undulating body motions to lift generation via oscillating airfoils.
Experimental data highlights this trend across different scales of flight. Insects, employing high-frequency flapping with rigid wings, operate at an Undulation number of approximately 0.24 ± 0.12. Bats, which utilize flexible airfoils capable of active cambering, show even lower values around 0.11 ± 0.06 (based on the data in [13]). Birds exhibit the lowest undulation numbers of all, at approximately 0.06 ± 0.02, relying on semi-rigid aerofoils to generate lift. Despite these differences in morphology, efficient cruising across these groups is characterized by a Strouhal number (St = f A/V f ly ) generally falling between 0.2 and 0.4 [19].
Beyond kinematics, the interaction between the animal and the fluid serves two distinct biological functions: locomotion and pumping. Locomotion is defined as momentum transport, where the primary goal is thrust generation to transport the body through a stationary fluid. In this case, the body moves while the net fluid motion is minimized, as seen in fish swimming or birds flying. Conversely, pumping represents mass transport, where the body remains fixed (or effectively stationary) and the goal is to transport fluid past the structure. This function is critical for physiological processes such as feeding, respiration, and circulation, exemplified by the heart pumping blood or mussels filtering water [20].
To understand the dynamics of animals moving near surfaces, it is helpful to introduce several key non-dimensional numbers that characterize the interplay between inertial, gravi-tational, surface tension, and viscous for
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